Solving stochastic delay differential equation by using Runge-Kutta method and method of lines

In this study, we have examined a stochastic delay differential equation in one dimension. We employed the Runge-Kutta method along with the method of lines to address the issue, establishing stability analysis and studying the spread of discontinuities. Additionally, we derived a semi-discretization in the temporal domain. Subsequently, the Runge-Kutta method of order four and cubic Hermite Interpolation is derived and numerical examples are illustrated for the theoretical result.